Non-Enumerative Path Delay Fault Diagnosis

Transcription

1 Non-Enumerative Path Delay Fault Diagnosis Saravanan Padmanaban Spyros Tragoudas Department of Electrical and Computer Engineering Southern Illinois University Carbondale, IL 6901 Abstract The first non-enumerative framework for diagnosing path delay faults using zero suppressed binary decision diagrams is introduced. We show that fault free path delay faults with a validated non-robust test may together with fault free robustly tested faults be used to eliminate faults from the set of suspected faults. All operations are implemented by an implicit diagnosis tool based on the zero suppressed binary decision diagram. The proposed method is space and time non-enumerative as opposed to existing methods which are space and time enumerative. Experimental results on the ISCAS 85 benchmarks show that the proposed technique is on an average least three times more efficient to improve the diagnostic resolution than existing techniques. 1 Introduction With the advent of deep-submicron technology, testing for the performance of an integrated circuit (IC) has become a difficult task. Even small process variations can cause a fault in the circuit. Therefore before mass production of the IC, a small number (first silicon) is produced to perform the various tests and check for the performance of the IC. The check is performed by applying test vectors and comparing the expected output to the sampled output. We assume a slow-fast test application methodology on the combinational component of the digital synchronous circuit and that the circuit under diagnosis is functionally correct. The process of locating the region in the chip that caused the delay fault is termed as delay fault diagnosis. We consider the path delay fault (PDF) model for the purpose of diagnosis. The number of PDFs in a circuit can be exponential. This necessitates the need for a datastructure that can efficiently represent and manipulate the path delay faults. [8] introduced a methodology to represent path delay faults as zero-suppressed binary decision diagrams (ZBDD). The ZBDD is a canonical data structure and is a variant of the binary decision diagram and is very effective in representing and manipulating single PDF and multiple PDF. We use the ZBDD as the underlying data structure for the proposed diagnosis methodology. Different methods have been proposed for diagnosing GDFs and PDFs [9]. We propose an enhanced ZBDD based framework for path delay fault diagnosis based on effectcause analysis. In effect-cause analysis methods, a set of /03 $ IEEE input vector pairs are applied to the circuit under test, the sampled output of the applied test is compared against the expected logical output. This information is used to prune the set of possible faults, in an attempt to locate the fault. Each of the input vector pairs for which the expected and observed output is the same is termed as a passing test. Each of the input vector pairs for which the expected and observed output differ is termed as a failing test. The set of all passing tests (respectively failing tests) is termed as a passing set(respectively failing set). The set of all PDFs sensitized by the passing set and is guaranteed to be fault free is termed as a fault free set. The set of all PDFs sensitized by the failing set that could explain for the error observed is termed as a suspect set. Some PDFs sensitized by a passing test vector may not be indeed fault free. This happens if the PDFs are sensitized non-robustly, in which case the transition may be masked during the actual test application process. The PDF diagnosis methodology proposed in [9], defines the term fault free PDF as a PDF that is sensitized by a given passing test and is guaranteed to be fault free. These are precisely the single and multiple PDFs that are robustly tested by a passing test as a fault free PDF. We show here that an additional class of PDFs, those with a validatable non-robust(vnr) test in the passing set can also be be classified as fault free PDFs. A drawback of [9] is that it uses a graph (cyclic) based data structure to represent multiple path delay faults (MPDF). Each node in the graph is represents a SPDF (which can be exponential to the number of lines sensitized in the circuit). Thus the method is space enumerative to the number of single path delay faults (SPDF) since we have to explicitly store each SPDF as a node. However a MPDF is stored as a cycle in the graph. If a MPDF is to be removed from the set, it is done by removing the corresponding cycle from the graph. This removal process is enumerative, since each MPDF is removed one at a time. To overcome the usage of these complex data structures and graph operations, we introduce a ZBBD based method to non-enumeratively store and manipulate PDFs (SPDFs and MPDFs). Several PDFs in the suspect set may be fault-free. Based on the PDFs in the fault free and suspect set, many such PDFs in the suspect are shown to be fault free, thus reducing the search space of the potential fault. We define the resolution of the diagnosis process (diagnostic resolution) to be the ratio of the cardinality of the suspect set after all applied reductions to the cardinality of the original suspect

2 (a)passed Test - Pattern I (b)passed Test - Pattern II (c)failed Test - Pattern III Figure 1: Path Delay Fault Diagnosis - A Review Passing Test - T 1 and T Failing Test - T 3 PDF ID PDF Sensitization Type PDF ID Suspect Set Type PD 1 "f::::0:1g Robust SPDF FD 1 "f::::0:1g SPDF PD "f::::1g Non-Robust SPDF FD #f:::0:1g SPDF PD 3 "f::::0:1g Non-Robust SPDF FD 3 fp 1:::1, MPDF P 3::::1g # Table 1: Sensitized Path Delay Faults set. This is the reduction in the cardinality of the suspect set expressed as a ratio. Section illustrates improvement in resolution when using PDFs with VNR tests. It also summarizes the rules for improving the diagnostic resolution, which is a clear super set of those introduced in [9]. Section 3 shows the data structure and algorithms that non-enumeratively improve the diagnostic resolution. It also presents the first nonenumerative algorithm to find the exact set of PDFs with a VNR test for a given passing set. Section 4 presents our diagnosis methodology using the procedures and operators introduced in Sections and 3. Experimental results and conclusion are provided in Sections 5 and 6 respectively. Diagnosis using PDFs With a VNR Test When a PDF P is shown to be fault free, then any PDF of higher cardinality which is a superset of P cannot have a delay fault [4]. The conventional effect-cause analysis based method prunes the suspect set by eliminating PDFs in the suspect set whose subfault is a robustly tested PDF in the fault free set. By the definition of a robustly tested path delay fault, some PDFs in the suspect set could not have caused the fault at the output. A class of PDFs termed as Path Delay Faults with VNR test has been defined in [10] and has been shown to have the quality of a robust fault. Definition 1 (Validatable Non-Robust Test) A set of two pattern tests S is termed as a validatable non-robust test for a path P if and only if no element of S is a robust test for P and if the circuit under test passes all tests in S, it can be concluded that the desired transition propagates along the path P in the time allowed. (Restated from [10]) A non-robust test for a PDF becomes invalid if the signal arriving at any of the non-robust off-inputs has a delay fault. If the PDF through the non-robust off-inputs can be tested robustly and is fault free, the non-robust test for the target PDF, together with the robust test for the PDFs through the non-robust off-inputs, form a validatable non-robust (VNR) test for the target PDF. A VNR test is guaranteed to detect a delay fault on the target PDF independent of the delays in the rest of the circuit if the circuit passes all the tests. For a given instance of a circuit shown in Figure 1, let the diagnostic test set T consist of three test vectors T 1 = f10001; 10100g, T = f01001; 10100g and T 3 = f01010; 11100g. Assume that the vectors T 1 and T are passing tests and T 3 be a failing test. Figures 1a, 1b and 1c show the PDFs sensitized by the test vectors T 1, T and T 3 respectively. Table 1 shows the type of each PDF (single or multiple) and its corresponding sensitization type 1. Based on the definition of a PDF with VNR test, tests T 1 and T form a VNR test for the PDF PD 3. (The non-robust off-input of is the line on PD 1 ). The test guarantees that the PDF PD 3 is fault free. Our method prunes the suspect set using the set of PDFs tested by robust and VNR tests from the passing set. Namely, the suspect set ffd 1 ;FD ;FD 3 g can be pruned using the set fpd 1 ;PD 3 g instead of just using the fault free PDF fpd 1 g that has been tested robustly by passing test T 1. The PDF FD 1 can be eliminated from the suspect set, pruning the suspect set to ffd ;FD 3 g. Without using the PDFs with a VNR test no pruning of the suspect set is possible. The definition of a PDF with a VNR test is recursive and this causes computation difficulties. However Section 3.1 presents a method where PDFs with VNR tests of any cardinality can be identified non-enumeratively by only three traversals on the passing test set. We emphasize that VNR tests may sometimes be invalid for PDF testing. However they can be used for in diagnosis without any skepticism. For more detailed discussions see also [5]. The remaining of the section gives the set of rules for performing diagnosis. Let the test set T consists of X 1 Table 1 represents the SPDFs as its constituent path together with the transition on its primary input. The MPDF is represented by its constituent subpaths together with the transition on its primary output

3 vectors - ft 1 ;T ;:::;T X g, with m passing tests and n failing tests. Let P = fp 1 [ P :::P m g be the set of fault free PDFs tested by each of the m passing tests. Let F = ff 1 [ F :::F n g be the set of PDFs tested by each of the n failing test vectors in T. Once the fault free set and the suspect set are derived, the process of diagnostic resolution can be performed using the following rules: 1. Let Q i be a SPDF in the fault free set P tested by a robust test or a VNR test. Any superset of Q i in the suspect set F (such as MPDF - Q i Q j ) can be eliminated. We call such MPDFs as redundant PDFs. This elimination is valid because, a MPDF can have a delay fault only if all subfaults of the MPDF have a delay fault.. Let Q i Q j be a MPDF in the fault free set P tested by a robust test or a VNR test. Any superset of Q i Q j in the suspect set F (such as MPDF - Q i Q j Q k ) can be eliminated from the suspect set. We note that [9] only uses a part of the rules shown, taking advantage of the fault free PDFs with robust tests alone. In addition to the above said rules, the set of fault free PDFs can be optimized for computational purposes. Let Q i and Q i Q j be a SPDF and MPDF in the fault free set P, respectively. Then Q i Q j can be eliminated from the set P because, if the SPDF Q i is fault free, then Q i Q j or any other MPDF of which Q i is a subfault, is also guaranteed to be fault free. Although such PDF elimination does not improve the resolution, it is very important for computational purposes. 3 Data Structures and Operations The process of extracting a fault free set is performed in two phases. The first phase involves the identification of the robustly tested PDFs and the second phase involves the identification of PDFs with VNR tests. For a given passing set, the set of tested PDFs needs to be extracted nonenumeratively and stored in a compact format. It has been shown in [8] that the problem of representing PDFs can be reduced to representing them as combinational sets. This task can be performed by just using standard ZBDD operators introduced in [7]. Extraction of the fault free set is performed by three passes on the passing test set. During the first pass, we extract all PDFs (single and multiple) that are robustly tested by the passing set. While performing the first pass, we collect all the information regarding fault free PDFs that are robustly tested by the passing set. The second pass identifies, the set of PDFs that are non-robustly tested by the passing set. During the third pass, we identify the set of PDFs with VNR which is the subset of the non-robustly tested PDFs (identified during the second pass) for which the off-inputs have a passing robust test. The example below briefly illustrates the methodology to extract the set of robustly tested PDFs for a given test vector. s1 6 s s1 s s (a) Simulation of T 11 9 T (b) PDFs Sensitized Figure : Extraction of the Sensitized PDFs Let the test vector T = f11101; 11110g be a passing test applied to the circuit shown in Figure a. Each line and gate of a circuit be assigned a unique variable and each of the primary input is assigned with two variables (one for rising and one for falling transitions). The presence of a variable v in a combinational element representing a PDF implies that v =1and the absence implies v =0. The numbers marked on the lines are the variables used to denote that line. For the primary inputs, variables 1-5 are used to represent the rising transitions on each input and variables 18- are used to represent the falling transitions. The bold lines in the circuit in Figure a indicate the sensitized lines. Procedure: Extract RPDF for each test t in test set T do for each gate g in topological order do if g is a primary input then assign g a variable depending on the transition else if any line l of g is robustly sensitized then store partial PDFs in set R l else if g is robustly co-sensitized then store partial PDFs in set R l for each primary output line l do R t ψ R t [ R l Eliminate redundant PDFs from R T and R t R T ψ R T [ R t The PDFs tested robustly by a single passing test can be identified by a single topological traversal. At each gate, all possible partial PDFs from the primary input to that gate are stored as a set. In the circuit shown in Figure a, the partial PDFs from primary inputs to line 1 and 13 are R 1 = f4:9:13g and R 13 = f:1g respectively. Gate 9 is co-sensitized, so a product operation is performed between the two partial products to represent the co-sensitization between them. At the end of the topological traversal each primary output contains the set of robustly tested PDFs from all the primary inputs to that particular primary output. The PDFs tested by the passing test t is R t =f4:9:11:16; 4:9:1:13:17:g, is represented as a ZBDD as shown in Figure b. The set of PDFs tested robustly by the passing set T, R T is obtained by performing a union of the sets of PDFs tested robustly by each test t in the set T. This method is formally outlined by Procedure Extract RPDF. Procedure: Eliminate(P, Q) Ensure: Q 6= ; Result ψ P (P (Q Λ (P ff Q))) return (Result) We now turn our attention to implement a procedure to eliminate redundant PDFs, which in fact can also be used to improve the diagnostic resolution. In [8] a new ZBDD operator, the containment operator (ff) was introduced. The

4 (a) CircuitUnder Diagnosis (b)passed Test - Pattern I (c)passed Test - Pattern II Figure 3: Identification of PDFs with a VNR Test R T =f" g N 1 = ; ; N = f" ," g N l N l Line R l Before VNR Check After VNR Check T P l Line R l Before VNR Check After VNR Check T P l 3 - " , " , " " " " , , " " " , " " "4 10 Table : Procedure to Extract PDFs tested by VNR tests operator can be used together with other ZBDD operators to identify if a minterm is contained in another minterm. Definition (Containment) Containment (P ff Q), is the union of all the quotients of dividing P by the cubes of Q. Example: Let P = fabd, abe, abg, cde, ceg, eghg and Q = fab, ceg. Then (P ff Q) is obtained as: (P ff Q) = (P /fabg) [ (P /fceg) = fd, e, gg[fd, gg = fd, e, gg More details about the implementation of the containment operator can be found in [8]. A new procedure Eliminate (described in Procedure Eliminate()) is introduced, that uses the containment operator and other ZBDD based set operators to determine whether minterm P is contained in minterm Q. Such a procedure is central for the diagnosis problem studied here. Procedure Eliminate() shows that the Eliminate procedure uses the standard ZBDD operators, which clearly is a measure for the time efficiency of the operator. In the remaining, we illustrate the use of the Eliminate procedure in the diagnosis problem studied in this paper. Example: Assume X 1 = fabd, abe, abg, cde, ceg, eghg be the set of MPDFs and X = fab, ceg be the set of SPDFs. Then a call to the procedure Eliminate(X 1 ;X ) results in the set feghg which contains MPDFs that does not contain any of the SPDFs of X.IfX 1 is described as a set of PDFs and X as a set of subfaults, then Eliminate(X 1 ;X ) identifies the set of PDFs in X 1 which does not contain any subfaults of X. This procedure can be used to eliminate the fault free MPDFs from the suspect set (similar to X 1 ), using the fault free set (similar to X ). 3.1 Identifying PDFs with a VNR Test In this paper we present the first non-enumerative method to identify the exact set of PDFs with VNR test, of any cardinality. Let R T represent the set of all PDFs robustly tested by the passing set T. For a given line l, let RT l represent the set of all partial PDFs tested robustly by the passing set T, originating from the line l and terminating at any primary output. This is a subset of set R T. For a given passing test t T, let N t represent the set of all PDFs non-robustly tested by a passing test t. For a given line l, let Nt l represent the set of all partial PDFs tested non-robustly by a passing test t, originating from the line l and terminating at any primary output. For a given line l, let Pt l represent the set of all partial PDFs tested robustly by a passing test t, originating from the line l and terminating at any primary output. Procedure: Extract VNRPDF R T ψ Set of all PDFs tested by robust tests from passing set T for each line l in topological order do R l T ψ partial PDFs tested robustly from l to Primary Outputs for all t such that t passing set T do N t ψ Set of all PDFs tested non-robustly by test t for all l such that l Primary Input do Nt l ψ partial PDFs tested non-robustly from l to Primary Outputs for each line l in topological order; l= Primary Input do Pt l ψ partial PDFs tested robustly from Primary Inputs to l Nt l ψ Subset of Nlp t from l to Primary Outputs (l p is the predecessor of l) if l is non-robustly sensitized then for each off-input l o of l do /*Check for Robustly Tested Off-Input*/ Nt l ψ (Nt l ((R T (R lo T Λ P t lo lo )) ff Pt )) if Nt l 6= ; then Pt l ψ Extend P lp t from l p to l for all l such that l Primary Output do VNRψ VNR[ Pt l The set R T, is computed by Procedure Extract RPDF. Then a topological traversal and for each line l in the circuit, we compute RT l. Subsequently, for each passing test t T, the set of all PDFs non-robustly tested by the passing test

5 t is done by a modified Procedure Extract RPDF where we only change the sensitization criteria. Next each line l in the circuit is processed in topological order. If a line l is robustly sensitized, Pt l is propagated to a line which is the successor of l. If line l is non-robustly tested, then a check is performed at each off-input l o of the gate whose on-input is l, to identify if there exists a VNR test for the partial PDFs terminating at line l. If there exists a VNR test at the off-inputs, then there exists no delay fault at any off-input l o and we propagate the set Pt l forward to a successor. For each primary output po, the set P po t represents the set of fault free PDFs with VNR tests that terminate at the output po. Let us now describe the method. Consider the CUD shown in Figure 3a (with line numbers indicated). Let the passing set be T = ft 1 ;T g where T 1 = f100001, g and T = f010001, g shown in Figures 3b and 3c respectively. The set of PDFs with robust tests from T is R T =f" g. The set of PDFs with VNR tests are identified using the Procedure Extract VNRPDF. The Table illustrates the procedure to extract PDFs tested by VNR tests. Test T 1 does not test any PDF non-robustly. The set of PDFs non-robustly tested by T is f" , " g. N 9, the set of partial PDFs tested by T non-robustly from line 9 to the primary output, is initialized to f , g. R T 15, the set of partial PDFs tested by robust tests of T, from line 15 to the primary output, is initialized to f0 1 4g. We observe that line 14 is non-robustly tested. A check for a VNR test is performed using the relation, N 14 ψ N 14 ((R T (RT 15 Λ P )) ff P ) where N 14 = f , g before VNR check, 15 = f4 10g and RT = f g. So N becomes f g after the VNR check. The partial path f g allows the PDFs of 14 to propagate without a delay fault despite line 14 being non-robustly tested. The off-input (line 15) is robustly tested. At the end of topological traversal, P 4 = f" g. It can be concluded that the PDF f" g is a PDF with VNR test and is fault free. 4 Path Delay Fault Diagnosis For a given passing set T, procedures Extract RPDFs and Extract VNRPDF extracts the fault free set. For the failing set F, a procedure (a modified version of the procedure Extract RPDF) is used to identify all sensitized PDFs. We first extract the set of sensitized PDFs for the passing set and then the failing set. Using the Eliminate operator and other standard ZBDD operators, we propose a method to perform the diagnosis using the rules provided in Section. Let P m represent the fault free MPDFs (robustly or VNR tested). Let P s represent the fault free SPDFs (robustly or VNR tested). Let S be the suspect set representing the set of suspicious PDFs. The whole process of diagnosis is described as shown below: 1. Phase I - Extract the sets P s,p m and S.. Phase II - Optimize the fault free set by eliminating the redundant MPDFs in the set. For example, the PDF P i P j is eliminated from the fault free set if PDF P i is also present. 3. Phase III - Eliminate the PDFs from the suspect set, that are present in both the suspect set and the fault free PDFs. Subsequently, eliminate the fault free MPDFs from the suspect set using the optimized fault free PDFs. Procedure: Diagnosis S =(S P s) S =(S P m) S = Eliminate(S, P s) S = Eliminate(S, P m) Note that the Procedure Eliminate does not eliminate the PDFs that are common to both the suspect set and the fault free set. PDFs common to the suspect set and the fault free set are eliminated using a set difference operator. Then we eliminate the fault free MPDFs from the S using the set of optimized fault free PDFs. More formally the Phase III is described as the Procedure Diagnosis. 5 Experimental Results The experiments were run on a 750MHz SUN Blade-1000 workstation with 1GB RAM. The ISCAS 85 benchmarks were used as the circuits under diagnosis. For the purpose of experimentation, tests were generated using the method proposed in [6]. The method of [6] generates robust and non-robust tests. However it does not generate pseudo- VNR tests. Hence the test sets generated for the ISCAS 85 benchmarks consist only of robust and non-robust tests. Out of the tests generated, 75 tests were assumed to form the failing set and the rest be the passing set. Table 3 reports the result of extracting the fault free PDFs and the optimization of the set of fault free PDFs. Column shows the cardinality of the passing set. Columns 3 and 4 show the number of fault free MPDFs and SPDFs tested by the given test set, respectively. Column 5 reports the number of fault free MPDFs after optimization (i.e., elimination) using the set of fault free PDFs with robust tests (SPDFs and MPDFs). After the fault free PDFs with robust tests are extracted and optimized, the set of PDFs with VNR tests is extracted. The number of PDFs with VNR tests is shown in Column 6. Certain MPDFs in the fault free set can be still optimized using the identified fault free PDFs with VNR tests (reported in Column 7). The cardinality of the fault free set of PDFs is reported in Column 8 (Sum of Columns 4,6 and 7). Column 9 shows the total processing time. The performance of a diagnosis methodology should not be judged by the total processing time taken. Time is not a crucial factor in fault diagnosis. For a given passing set, the more the number of faults identified as fault free, the better the chance for locating the fault. The proposed method is guaranteed to identify at least as many fault free PDFs as the methodology of [9]. The increase in the number of the fault free PDFs is attributed to the PDFs with the VNR tests. More details about the fault free set is shown in Table 4. Column shows the number of PDFs identified

6 Benchmark Passing Test Fault Free Fault Free MPDFs PDFs with MPDFs Fault Free Time Vectors MPDFs SPDFs (Optm.) VNR Test (Optm.) PDFs (sec) C880, , , , C ,697 1,09 16, , , C ,404,693 3,35,693 4,648,35 30, C670 5,353 3,634 9,449 3,15 1,809 3,089 14, C ,730 5,47,33 4,861 6,91 4,814 34, C ,373 4,06 33,936,961 4,613,961 41, C688 3, , , C755 7,90 6,977 5,577 6,87 4,931 6,863 64, Table 3: Identification of Fault Free PDFs Benchmark Fault Free PDFs Fault Free PDFs Increase in [11] (proposed) Fault Free PDFs C880 9,583 10,71 1,19 C ,698 41,377 3,679 C1908 6,018 30,35 4,307 C670 1,601 14,347 1,746 C3540 7,193 34,058 6,865 C ,897 41,510 4,613 C688,968 3, C755 59,449 64,371 4,9 Table 4: Improvement in Diagnosis Suspect Set Diagnosis [11] Diagnosis (proposed) Resolution Benchmark MPDFs SPDFs Cardinality MPDFs SPDFs Cardinality MPDFs SPDFs Cardinality [11] (proposed) Improvement C % C % C % C % C % C % C % C % Table 5: Result of Diagnosis as fault free using the method in [9]. (Sum of Columns 4 and 5). Column 3 reports the number of PDFs identified as fault free by the proposed method and Column 4 reports the increase in the number of fault free PDFs. The results of the diagnostic resolution process is presented in Table 5. Columns and 3 represent the number of suspicious MPDFs and SPDFs in the suspect set. Column 4 represents the cardinality of the suspect set, which is the sum of Columns and 3. Columns 5 and 6 show the size of the MPDFs and SPDFs in the suspect set after performing diagnosis using the set of fault free PDFs with robust tests as proposed in [9] (The set shown in Column of Table 4). The cardinality of the resulting suspect set is shown in Column 7. Columns 8 and 9 show the size of the MPDFs and SPDFs in the suspect set after performing diagnosis using the set of fault free PDFs with robust and VNR tests, as proposed in this paper. The cardinality of the resulting suspect set using the proposed method is shown in Column 10. It can be seen from Column 10 that the size for the suspect set is greatly reduced when compared to [9] which is attributed to the identification of more fault free PDFs. Columns 11 and 1 shows the resolution of the diagnosis process using the method of [9] and the proposed technique. It is observed from Column 13 that there is an average increase of 360% in the resolution using the proposed method over [9]. The method [9] presented results on some of the IS- CAS 89 benchmarks and showed that the method could result in a reasonable resolution. However Table 5 (Column 11) shows that the method results in a poor resolution of 10%. This is attributed to the fact that the ISCAS 89 benchmarks that were examined in [9] have more than 90% of the PDFs to be robustly testable (shown by [1]). However the ISCAS 85 benchmarks for which results are reported in Table 5 have less than 15% of the PDFs in the circuit to be robustly testable (shown by [3]). This impacts the performance of [9]. The proposed method is expected to perform better if the test set generated for performing diagnosis, explicitly targets the generation of pseudo-vnr tests, like []. 6 Conclusion A new non-enumerative framework has been introduced for performing diagnosis using polynomial number of ZBDD operations. The method also takes advantage of the PDFs with VNR tests to improve diagnostic resolution by three folds than existing work. The first method to identify PDFs with VNR test for a given test set is also introduced. The proposed algorithms use the ZBDD as the underlying data structure which has been shown to be effective in storing and manipulating PDFs, non-enumeratively. 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